But what about Vy? Why can't we say that for the best Vy we need the greatest excess thrust? Okay, power = thrust x time, and with Vy time is an important factor. But: thrust = power/time. Here we also consider the time.
As Italia has said, your equations (power = thrust x time) and (thrust = power / time) are both incorrect.
To understand when best ROC occurs when excess power is greatest it might be better to use the equations below.
Power = work done / time taken to do that work.
And
Power = force x velocity
In a climb work must be done to achieve the following two things:
1. Move the aircraft forward at whatever TAS we have chosen. To achieve this we must exert a thrust force that is equal to the drag. So the power used to carry out this part of the work is equal to drag x TAS. This is the POWER REQUIRED simply to maintain our TAS.
2. Move the aircraft vertically upwards at whatever vertical speed (ROC) we have chosen. To achieve this we must apply a vertical component of thrust that is equal to the weight of the aircraft. So the power that is required to achieve this = Weight x ROC.
The power in paragraph 2 (Weight x ROC) is the extra power (or excess power) that we need to achieve the selected ROC. If our maximum power available is just equal to the power required (in paragraph 1) , we can maintain our selected TAS, but we have no extra power available to achieve any ROC. So to achieve the maximum possible ROC we must fly at the speed at which we have the maximum amount of excess power available.